This question is a tad niche, hopefully it can get an answer. I've been trying to digest the paper "Families of finite sets with minimum shadows"(https://link.springer.com/article/10.1007/BF02579261), but cannot see how they derive one of their steps.
To give some background, for a family of $k$-sets $\mathcal{F}\subseteq [n]^{(k)}$, we say its $g$-shadow ($g<k$) is the family of sets $\Delta_g\mathcal{F}:=\{A\subseteq [n]:|A|=g, \exists F\in \mathcal{F}. A\subseteq F\}$. The Kruskal-Katona theorem says that given a family of sets $\mathcal{F}\subseteq[n]^{(k)}$ with $|\mathcal{F}|$ sets, its $g$-shadow is at least as large as the $g$-shadow of the initial segment of length $|\mathcal{F}|$ of $\mathbb{N}^{(k)}$ in the colexicographic (also called antilexicographic) order. In other words, initial colex segment families minimise shadows.
However, initial colex segment families are not always the unique set families that minimise shadow. This paper tries to characterise these extremal families.
The bit where I am stuck is that I do not see how equation (11) in the proof of Theorem 2.1 follows.
It seems to me that to obtain $\Delta_g\bar{\mathcal{A}}\supseteq\Delta_g\bar{\mathcal{B}}$ what we really want is $|\Delta_{k-1}\bar{\mathcal{A}}|\geq |\bar{\mathcal{B}}|$. While we have that $|\bar{\mathcal{B}}|=|\mathcal{B}|$, it is not clear to me that $|\Delta_{k-1}\bar{\mathcal{A}}|=|\Delta_{k-1}\mathcal{A}|$.